Astronomical Dating and the Internal Chronology of the Pentateuch

Some of the narratives in the Pentateuch can be associated with known astronomical events to provide absolute dates for biblical chronology.Comment: 29 pages, 12 figure

Game Networks

We introduce Game networks (G nets), a novel representation for multi-agent decision problems. Compared to other game-theoretic representations, such as strategic or extensive forms, G nets are more structured and more compact; more fundamentally, G nets constitute a computationally advantageous framework for strategic inference, as both probability and utility independencies are captured in the structure of the network and can be exploited in order to simplify the inference process. An important aspect of multi-agent reasoning is the identification of some or all of the strategic equilibria in a game; we present original convergence methods for strategic equilibrium which can take advantage of strategic separabilities in the G net structure in order to simplify the computations. Specifically, we describe a method which identifies a unique equilibrium as a function of the game payoffs, and one which identifies all equilibria.Comment: Appears in Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI2000

Correlated Equilibria of Classical Strategic Games with Quantum Signals

Correlated equilibria are sometimes more efficient than the Nash equilibria of a game without signals. We investigate whether the availability of quantum signals in the context of a classical strategic game may allow the players to achieve even better efficiency than in any correlated equilibrium with classical signals, and find the answer to be positive.strategic games, quantum mechanics, correlated equilibrium, coordination, entanglement, efficiency

Expected Utility Networks

We introduce a new class of graphical representations, expected utility networks (EUNs), and discuss some of its properties and potential applications to artificial intelligence and economic theory. In EUNs not only probabilities, but also utilities enjoy a modular representation. EUNs are undirected graphs with two types of arc, representing probability and utility dependencies respectively. The representation of utilities is based on a novel notion of conditional utility independence, which we introduce and discuss in the context of other existing proposals. Just as probabilistic inference involves the computation of conditional probabilities, strategic inference involves the computation of conditional expected utilities for alternative plans of action. We define a new notion of conditional expected utility (EU) independence, and show that in EUNs node separation with respect to the probability and utility subgraphs implies conditional EU independence.Comment: Appears in Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence (UAI1999

Team Decision Problems with Classical and Quantum Signals

We study team decision problems where communication is not possible, but coordination among team members can be realized via signals in a shared environment. We consider a variety of decision problems that differ in what team members know about one another's actions and knowledge. For each type of decision problem, we investigate how different assumptions on the available signals affect team performance. Specifically, we consider the cases of perfectly correlated, i.i.d., and exchangeable classical signals, as well as the case of quantum signals. We find that, whereas in perfect-recall trees (Kuhn [1950], [1953]) no type of signal improves performance, in imperfect-recall trees quantum signals may bring an improvement. Isbell [1957] proved that in non-Kuhn trees, classical i.i.d. signals may improve performance. We show that further improvement may be possible by use of classical exchangeable or quantum signals. We include an example of the effect of quantum signals in the context of high-frequency trading.Comment: 18 pages, 16 figure

Deriving the Qubit from Entropy Principles

The Heisenberg uncertainty principle is one of the most famous features of quantum mechanics. However, the non-determinism implied by the Heisenberg uncertainty principle --- together with other prominent aspects of quantum mechanics such as superposition, entanglement, and nonlocality --- poses deep puzzles about the underlying physical reality, even while these same features are at the heart of exciting developments such as quantum cryptography, algorithms, and computing. These puzzles might be resolved if the mathematical structure of quantum mechanics were built up from physically interpretable axioms, but it is not. We propose three physically-based axioms which together characterize the simplest quantum system, namely the qubit. Our starting point is the class of all no-signaling theories. Each such theory can be regarded as a family of empirical models, and we proceed to associate entropies, i.e., measures of information, with these models. To do this, we move to phase space and impose the condition that entropies are real-valued. This requirement, which we call the Information Reality Principle, arises because in order to represent all no-signaling theories (including quantum mechanics itself) in phase space, it is necessary to allow negative probabilities (Wigner [1932]). Our second and third principles take two important features of quantum mechanics and turn them into deliberately chosen physical axioms. One axiom is an Uncertainty Principle, stated in terms of entropy. The other axiom is an Unbiasedness Principle, which requires that whenever there is complete certainty about the outcome of a measurement in one of three mutually orthogonal directions, there must be maximal uncertainty about the outcomes in each of the two other directions.Comment: 8 pages, 3 figure

R\'enyi Entropy, Signed Probabilities, and the Qubit

The states of the qubit, the basic unit of quantum information, are $2\times2$ positive semi-definite Hermitian matrices with trace $1$. We characterize these states in terms of an entropic uncertainty principle formulated on an eight-point phase space.Comment: 11 pages, 1 figur

Correlated Equilibria of Classical Strategic Games with Quantum Signals

Correlated equilibria are sometimes more efficient than the Nash equilibria of a game without signals. We investigate whether the availability of quantum signals in the context of a classical strategic game may allow the players to achieve even better efficiency than in any correlated equilibrium with classical signals, and find the answer to be positive.Comment: 8 pages, LaTe

Projective Expected Utility

Motivated by several classic decision-theoretic paradoxes, and by analogies with the paradoxes which in physics motivated the development of quantum mechanics, we introduce a projective generalization of expected utility along the lines of the quantum-mechanical generalization of probability theory. The resulting decision theory accommodates the dominant paradoxes, while retaining significant simplicity and tractability. In particular, every finite game within this larger class of preferences still has an equilibrium.Comment: 7 pages, to appear in the Proceedings of Quantum Interaction 200